User:Tohline/2DStructure/AxisymmetricInstabilities
Axisymmetric Instabilities to Avoid
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When constructing rotating equilibrium configurations that obey a barotropic equation of state, keep in mind that certain physical variable profiles should be avoided because they will lead to structures that are unstable toward the dynamical development of shapedistorting or convectivetype motions. Here are a few wellknown examples.
RayleighTaylor Instability
Referencing both p. 101 of [ Shu92 ], and volume I, p. 410 of [ P00 ], a RayleighTaylor instability is a bouyancydriven instability that arises when a heavy fluid rests on top of a light fluid in an effective gravitational field, <math>~\vec{g}</math>. In the simplest case of spherically symmetric, selfgravitating configurations, the condition for stability against a RayleighTaylor instability may be written as,
<math>~( \vec{g} ) \cdot \nabla\rho</math> 
<math>~< </math> 
<math>~0 </math> 
[stable] , 
that is to say, the mass density must decrease outward. In an expanded discussion, [ P00 ] — see pp. 410  413 — derives a dispersion relation that describes the development of the RayleighTaylor instability in a planeparallel fluid layer that initially contains a discontinuous jump/drop in the density.
In addition, [ P00 ] — see pp. 413  416 — and [ Shu92 ] — see pp. 101  105 — both discuss circumstances that can give rise to the socalled KelvinHelmholtz instability. It is another common twofluid instability, but one that depends on the existence of transverse velocity flows and that is independent of the gravitational field.
PoincaréWavre Theorem
As [ T78 ] points out — see his pp. 78  81 — Poincaré and Wavre were the first to, effectively, prove the following theorem:
For rotating, selfgravitating configurations "any of the following statements implies the three others: (i) the angular velocity is a constant over cylinders centered about the axis of rotation, (ii) the effective gravity can be derived from a potential, (iii) the effective gravity is normal to the isopycnic surfaces, (iv) the isobaric and isopycnicsurfaces coincide." 
Among other things, this implies that for rotating barotropic configurations not only is the equation of state given by a function of the form, <math>~P = P(\rho)</math>, but it must also be true that,
<math>~\frac{\partial \dot\varphi}{\partial z}</math> 
<math>~=</math> 
<math>~0 \, .</math> 
[ T78 ], §4.3, Eq. (30) 
NOTE: We should investigate how this theorem comes into play in the context of our accompanying discussion of Type 1 Riemann ellipsoids. These are equilibrium triaxial, uniformdensity configurations in which the system's internal vorticity vector does not align with the tumbleaxis of the ellipsoid and, therefore apparently <math>~\dot\varphi</math> is not independent of <math>~z</math>.
Høiland Criterion
"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by Høiland (1941) (see also Ledoux's Chapter 10, pp. 499574 of Stellar Structure (1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however." 
— Drawn from p. 475 of N. R. Lebovitz (1967), ARAA, 5, 465 
As is stated on p. 166 of [ T78 ], in rotating barotropic configurations, axisymmetric stability requires the simultaneous satisfaction of the following pair of conditions:
<math>~\biggl(\frac{1}{\varpi^3} \biggr) \frac{\partial j^2}{\partial \varpi} + \frac{1}{c_P} \biggl( \frac{\gamma  1}{\Gamma_3  1}\biggr) ( \vec{g} ) \cdot \nabla s</math> 
<math>~></math> 
<math>~0 </math> 
[stable] ; 
[ T78 ], §7.3, Eq. (41)


<math>~g_z \biggl[ \frac{\partial j^2}{\partial \varpi} \biggl(\frac{\partial s}{\partial z} \biggr)  \frac{\partial j^2}{\partial z} \biggl(\frac{\partial s}{\partial \varpi} \biggr)\biggr]</math> 
<math>~></math> 
<math>~0 </math> 
[stable] . 
[ T78 ], §7.3, Eq. (42)

where, <math>~s</math>, is the local specific entropy, and <math>~j \equiv \dot\varphi \varpi^2</math>, is the local specific angular momentum of the fluid. According to [ T78 ] — see p. 168 — this pair of mathematically expressed conditions has the following meaning:
"A baroclinic star in permanent rotation is dynamically stable with respect to axisymmetric motions if and only if the two following conditions are satisfied: (i) the entropy per unit mass, <math>~s</math>, never decreases outward, and (ii) on each surface <math>~s</math> = constant, the angular momentum per unit mass, <math>~j</math>, increases as we move from the poles to the equator." 
Schwarzschild Criterion
In the case of nonrotating equilibrium configurations, the Høiland Criterion reduces to the Schwarzschild criterion. That is, thermal convection arises when the condition,
<math>~( \vec{g} ) \cdot \nabla s</math> 
<math>~> </math> 
<math>~0 </math> 
[stable] , 
[ T78 ], §7.3, Eq. (43)

is violated. This means that, in order for a spherical system to be stable against dynamical convective motions, the specific entropy must increase outward.
Solberg/Rayleigh Criterion
In the case of an homentropic equilibrium configuration, the Høiland Criterion reduces to the Solberg criterion. That is, an axisymmetric exchange of fluid "rings" will occur on a dynamical time scale if the condition,
<math>~\frac{dj^2}{d\varpi} </math> 
<math>~> </math> 
<math>~0 </math> 
[stable] , 
[ T78 ], §7.3, Eq. (44)

is violated. This means that, for stability, the specific angular momentum must necessarily increase outward. As [ T78 ] points out, this "Solberg criterion generalizes to homentropic bodies the wellknown Rayleigh (1917) criterion for an inviscid, incompressible fluid."
Modeling Implications and Advice
Here are some recommendations to keep in mind as you attempt to construct equilibrium models of selfgravitating astrophysical fluids.
RayleighTaylor instability: In order to avoid constructing configurations that are subject to the RayleighTaylor instability, be sure that lower density material is never placed "beneath" higher density material. For example …
 When building a bipolytropic configuration, a value for the (imposed) discontinuous jump in the meanmolecular weight will need to be specified at the interface between the envelope and the core. If you choose a ratio, <math>~{\bar\mu}_e/{\bar\mu}_c</math>, that is greater than unity, the resulting equilibrium model will exhibit a discontinuous density jump that makes the density higher at the base of the envelope than it is at the surface of the core. The core/envelope interface of this configuration will be unstable to the RayleighTaylor instability, but you won't know that until and unless you examine the hydrodynamic stability of the configuration.
Schwarzschild criterion: In order to avoid constructing configurations that violate the Schwarzschild criterion, be sure that the specific entropy of the fluid is uniform (marginally stable) or increases outward throughout the equilibrium structure, where the word "outward" is only meaningful when referenced against the direction that the effective gravity points. For example …
 Suppose that you build a spherically symmetric polytropic configuration whose structural index is, <math>~n</math>, but then you want to test the stability of the configuration assuming that compressions/expansions of individual fluid elements occur along adiabats for which the adiabatic index is, <math>~\gamma \ne (n+1)/n</math>. Although we generally think of polytropes as being homentropic configurations, if <math>~\gamma \ne (n+1)/n</math>, then different fluid elements will, in practice, evolve along adiabats that are characterized by different values of the specific entropy; that is, throughout the equilibrium model, <math>~\nabla s</math> will not be zero. Whether the specific entropy increase (stable) or decreases (unstable) outward will depend on whether you select a value for the evolutionary <math>~\gamma</math> that is greater than (stable) or less than (unstable) <math>~(n+1)/n</math>.
 According to Woosley's class lecture notes,
<math>~\frac{d\ln \rho}{d\ln P}\biggr_\mathrm{structure} = \frac{n}{n+1}</math>
<math>~></math>
<math>~\frac{1}{\gamma_g}</math>
<math>~\Rightarrow</math> stable
<math>~<</math>
<math>~\frac{1}{\gamma_g}</math>
<math>~\Rightarrow</math> unstable
This is another way of expressing the same stability criterion for polytopes.
 Examples: An n = 1 polytope is unstable toward convection if expansions (or contractions) occur along an adiabat with <math>~\gamma_g < 2</math>. Alternatively, an n = 5 polytope is unstable toward convection if expansions (or contractions) occur along an adiabat with <math>~\gamma_g < \tfrac{6}{5}</math>.
See Also
 Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148154) — On the Dynamics of Revolving Fluids
 P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584 597) — Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid (includes torus with nonuniform rotation)
 Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an equilibrium incompressible configuration must rotate uniformly on cylinders (the famous "PoincaréWavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"
 Referring to our accompanying discussion of Type 1 Riemann ellipsoids, it seems that uniform rotation on cylinders is not required. What's going on?
© 2014  2021 by Joel E. Tohline 